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Important continuous distributions

Solution.

1
E(T) = 400, k = , P(T ≥ 800) = 1 − P(T < 800) = 1 − F(800) =
400
800
= 1 − (1 − e − 480 ) = e −2 ≈ 0.135.

The gamma distribution. We may generalize the above discussion to obtain the PDF for the
interval between every r-th event in a Poisson process or, equivalently, the interval (waiting time)
before the r-th event. We begin by using the Poisson distribution to give

(kx) r−1
P(r − 1 events occur in interval x) = e −kx ,
(r − 1)!
from which we obtain
(kx) r−1
P(r − th event occurs in the interval [x, x + dx]) = e −kx kdx.
(r − 1)!

Thus the required PDF is
k
f(x) = (kx) r−1 −kx , (6.18)
e
(r − 1)!
which is known as the gamma distribution of order r with parameter k.

f(x)

1 λ = 1, r = 1
λ = 1, r = 2
λ = 1, r = 5
0.8
λ = 1, r = 10
0.6

0.4

0.2

0 x
0 2 4 6 8 10

Figure 6.8 – The PDF f(x) for the gamma distribution γ(λ, r) with λ = 1 and r = 1, 2, 5, 10.

Although our derivation applies only when r is a positive integer, the gamma distribution is
defined for all positive r by replacing (r − 1)! by Γ(r) in (6.18); see the appendix for a discussion
of the gamma function Γ(x). If a random variable X is described by a gamma distribution of order
r with parameter k, we write X ∼ γ(k, r); we note that the exponential distribution is the special
case γ(k, 1).
r r
E(X) = , Var(X) = . (6.19)
k k 2

The normal (Gaussian) distribution. A normal distribution is determined by the density:

1 (x−a) 2
f(x) = √ e − 2σ 2 , x ∈ R, (6.20)
σ 2π
in which a is the mean and σ is the standard deviation.
Consider some properties of the function f(x) given by the formula (6.20)

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